https://doi.org/10.1140/epjd/s10053-025-01062-2
Regular Article – Molecular Physics and Chemical Physics
The effect of space restriction on the bonding and vibrational properties of H
molecular ion
1
Université Paris-Saclay, CNRS, Institut des Sciences Moléculaires d’Orsay-ISMO (UMR 8214), 91405, Orsay Cedex, France
2
Theoretical and Computational Physics Section, Raja Ramanna Centre for Advanced Technology, 452013, Indore, India
3
Homi Bhabha National Institute, Training School Complex, Anushaktinagar, 400094, Mumbai, India
a
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Received:
26
May
2025
Accepted:
9
September
2025
Published online:
30
September
2025
Abstract
When the space surrounding an atom or molecule is restricted, its electron density is modified compared to its unrestricted counterpart, which further alters its physical and chemical properties. To model the effect of space restriction, or confinement, on atoms and molecules, often the hard-boundary condition is employed to solve the corresponding Schrödinger equation. In the present work, we analyze the modifications in a chemical bond when the space surrounding the molecule is restricted such that some part of the space is forbidden to the electrons. To this end, we choose the simplest hydrogen molecular ion H, in the presence of an external hard-sphere simulating the restriction of space available to the electrons due to Pauli exclusion by an atom close to the molecule. The hard-sphere is moved along the molecular axis and the axis transverse to it to study the changes in the ground-state energy, stretching frequency, kinetic, and potential energies of an H molecule. For this purpose, we employ a variational approach using a suitably constructed energy estimator that goes beyond the Born–Oppenheimer approximation. We find that as the sphere moves closer to the molecule along the molecular axis, the ground-state energy and the vibrational frequencies get enhanced. On the other hand, for the same case, the potential energy decreases, and the kinetic energy increases. When the sphere moves along the axis transverse to the molecular axis, the changes in the energy and vibrational frequencies show similar trends; however, the magnitude of the changes is significantly smaller than in the corresponding molecular axis case. For the transverse case, both potential and kinetic energies get enhanced marginally. These results demonstrate that the axial approach of the hard surface is more efficient in inducing a bond breaking of the H molecular ion. The shift of the electron bond by the external sphere explains these results.
© The Author(s) 2025
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